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%I #11 May 15 2017 18:25:11
%S 1,-1,-1,3,7,-25,-31,427,127,-12465,-2555,555731,1414477,-35135945,
%T -57337,2990414715,118518239,-329655706465,-5749691557,45692713833379,
%U 91546277357,-7777794952988025,-1792042792463,1595024111042171723,1982765468311237,-387863354088927172625
%N Numerator of Bernoulli(n, 1/4).
%C From _Wolfdieter Lang_, Apr 28 2017: (Start)
%C The rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A285061(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) define generalized Bernoulli numbers, named B[4,1](n), in terms of the generalized Stirling2 numbers S2[4,1]. The numerators of r(n) are a(n) and the denominators A141459(n). r(n) = B[4,1](n) = 4^n*B(n, 1/4) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383.
%C The generalized Bernoulli numbers B[4,3](n) = Sum_{k=0..n} ((-1)^k/(k+1))* A225467(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) satisfy
%C B[4,3](n) = 4^n*B(n, 3/4) = (-1)^n*B[4,1](n). They have numerators (-1)^n*a(n) and also denominators A141459(n). (End)
%H Vincenzo Librandi, <a href="/A157817/b157817.txt">Table of n, a(n) for n = 0..250</a>
%F From _Wolfdieter Lang_, Apr 28 2017: (Start)
%F a(n) = numerator(Bernoulli(n, 1/4)) with denominator A157818(n) (see the name).
%F a(n) = numerator(4^n*Bernoulli(n, 1/4)) with denominator A141459(n) = A157818(n)/4^n.
%F a(n)*(-1)^n = numerator(4^n*Bernoulli(n, 3/4)) with denominator A141459(n).
%F (End)
%t Table[Numerator[BernoulliB[n, 1/4]], {n, 0, 50}] (* _Vincenzo Librandi_, Mar 16 2014 *)
%Y For denominators see A157818 and A141459.
%K sign,easy,frac
%O 0,4
%A _N. J. A. Sloane_, Nov 08 2009
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